Cellular Automata
1
Introduction
 What are Cellular Automata?

CA are discrete dynamic systems.



CA's are said to be discrete because they operate
in finite space and time and with properties that
can have only a finite number of states.
CA's are said to be dynamic because they exhibit
dynamic behaviors.
Basic Idea: Simulate complex systems by
interaction of cells following easy rules.
2
Introduction
 To put it another way:

“Not to describe a complex system with
complex equations, but let the complexity
emerge by interaction of simple individuals
following simple rules.”
3
Cellular Automata
•A CA is a spatial lattice of N cells, each of which is one of k
states at time t.
•Each cell follows the same simple rule for updating its state.
•The cell's state s at time t+1 depends on its own state and the states of some
number of neighbouring cells at t.
• For one-dimensional CAs, the neighbourhood of a cell consists of the cell itself
and r neighbours on either side. Hence, k and r are the parameters of the CA.
•CAs are often described as discrete dynamical systems with the capability to
model various kinds of natural discrete or continuous dynamical systems
4
History
 Original concept of CA is most strongly
associated with John von Neumann.
 von Neumann was interested in the
connections between biology and the then
new study of automata theory.
 Stanislaw Ulam suggested that von Neumann
use a cellular automata as a framework for
researching these connections.
5
History
 The original concept of CA can be credited to
Ulam, while the early development of the
concept is credited to von Neumann.
 Although von Neumann made many
contributions and developments in CA, they
are commonly referred to as “non-von
Neumann style”, while the standard model of
computation (CPU, globally addressable
memory, serial processing) is know as “von
Neumann style”.
6
BACKGROUND
Time Frame
Early 50’s
Major Players
J. Von Neuman, E.F. Codd,
Henrie & Moore , H Yamada &
S. Amoroso
‘60s & ‘70s A. R. Smith , Hillis, Toffoli
Contribution
Modeling biological
systems - cellular models
Language recognizer, Image
Processing
‘80 s
S. Wolfram ,Crisp,Vichniac Discrete Lattice,statistical
systems, Physical systems
‘87 - ‘96
IIT KGP, Group
Additive CA,
characterization,applications
‘97 - ‘99
B.E.C Group
GF (2p) CA
7
Why should we study Cellular
Automata?
 Powerful computation engines.
 Discrete dynamical system simulators.
 Conceptual vehicle for exploring pattern
formation.
 Models of fundamental physics.
8
Why should we study Cellular
Automata?
 Powerful computation engines.

Allow very efficient parallel computation
 Discrete dynamical system simulator.

Allow for a systematic investigation of complex
phenomena.
 Original models of fundamental physics.

Instead of looking at the equations of
fundamental physics, consider modelling them
with CA.
9
Applications
 Simulations






Gas behavior
Forest fire propagation
Urban development
Traffic flow
Air flow
Crystallization process
 Alternative to differential equations
10
Components
 Cell



Basic element of a CA.
Cells can be thought of as memory elements
that store state information.
All cells are updated synchronously according
to the transition rules.
 Lattice


Spatial web of cells.
Simplest lattice is one dimensional.
11
Behavior
 Local interaction leads to global dynamics.


One can think of the behavior of a cellular
automata like that of a “wave” at a sports
event.
Each person reacts to the state of his
neighbors (if they stand, he stands).
12
Behavior
 Rule Application
 Next state of the core cell is related to the states of the
neighborhood cells and its current state.
 An example rule for a one dimensional CA: 011->x0x
 All possible states must be described.
 Next state of the core cell is only dependent upon the
sum of the states of the neighborhood cells.
 For example, if the sum of the adjacent cells is 4 the
state of the core cell is 1, in all other cases the state of
the core cell is 0.
13
Behavior
 Neighborhoods




von Neumann
Moore
Extended Moore
What about border cells?

Common approach is to “wrap” around.
14
Behavior
 Neighborhoods
 von Neumann – Composed of the cells above, below, left
and right. Has a radius of 1.
15
Behavior
 Neighborhoods
 Moore – Composed of the same cells as the von
Nuemann neighborhood, but also includes the diagonal
cells. Has a radius of 1.
16
Behavior
 Neighborhoods
 Extended Moore – Composed of the same cells as the
Moore neighborhood, but the radius of neighborhood is
increased. Has a radius of 2.
17
Behavior
 Rule Systems

Rule systems define the dynamic behavior of
the CA.

Even in a simple one dimensional system, there
are many different ways to determine the next
step.
 Could base the next state of the cell off of the sum of
the states of your neighbors (Game of Life).
 Could modify the scope of the neighborhood, so the
resulting neighbors could be local (touching), close
(neighbor’s neighbors) or global (anywhere in the
system) or possibly use random neighbors.
 Could allow the cells to grow and die.
18
Characteristics
 Discrete lattice of cells.
 Homogeneity – all of the cells of the lattice are
equivalent.
 Discrete states – each cell takes on one of a
finite number of possible discrete states.
 Local interactions – each cell interacts only
with cells that are in its local neighborhood.
19
Characteristics
 Discrete dynamics – at each discrete time unit,
each cell updates its current state according to
a transition rule taking into account the states
of the cells in its neighbourhood.
20
Example of 1-D cellular automaton
•For a binary input N long, are there more 1s than 0s?
•Set k=2 and r=1 with the following rule:
Cell + 2
neighbours:
000
001
010
011
100
101
110
111
0
0
0
1
0
1
1
1
Result:
That is, the value of a cell at time t+1 will depend on its value
and the values of its two immediate neighbours at time t.
This is a form of ‘majority voting’ between all three cells.
21
One-Dimensional CA Illustration
r
t
i - r ... i - 1 i i+1 ... i+r
hi
f
t+1
i
22
One-Dimensional Cellular Automata (CA)
 Consists of a linear array of identical cells
(called a lattice), each of which can be in a
finite number of k states.
 The (local) state of cell i at time t is denoted:
sti    {0,1,..., k  1}
 The (global) state st at time t is the
configuration of the entire array,
s t  ( st0 , st1 ,..., stN 1 )   N
 where N is the (possibly infinite) size of the
array.
23
One-Dimensional CA cont.
 At each time step, all cells in the array update their state
simultaneously, according to a local update rule f :h  s .
 This update rule takes as input the local neighborhood
configuration h of a cell.
 A local neighborhood configuration h consists of si and its 2r
nearest neighbors (r cells on either side):
hi  ( s i  r ,..., s,..., s i  r )
 r is called the radius of the CA.
 The local update rule f, which is the same for every cell in the
array, can be represented as a lookup table, which lists all
possible neighborhood configurations (| h |  k 2 r 1 )
sti1  f (h it )
24
rule 6
Ci-1(t) Ci(t)
Ci+1(t) Ci(t+1)
0
0
0
0
0
0
1
1
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
1
1
1
0
1
1
1
1
0
Wolfram rule
Sum to 0 --> 0
1 --> 1
2 --> 1
3 --> 0
0110 = ‘rule 6’
Flake: k(2r+1) rules (??)
(k-1)(2r+1) +1 works
25
Example One-Dimensional CA
Rule 110








The number of states, k=2.
The alphabet
  {0,1}   k
The size of the array, N=11.
The configuration space h
 N  {( 0,0,0,0,0,0,0,0,0,0,0), (0,0,0,0,0,0,0,0,0,0,11),...}
The radius r = 1.
i
i
The rule table : st 1  f (h t )
 neighborhood
: 000 001 010 011 100 101 110 111

f: 0 1 1 1 0 1 1 0
This is rule 110 (base 10) because the output states are:
01101110 (base 2). Read right to left. Known as Wolfram
notation.
Rule 110 supports universal computation.
26
Rule Table f :
Neighborhood: 000 001 010 011 100 101 110 111
0
1
1
1
0
1
1
0
Output bit:
Lattice:
Periodic
boundary
conditions
t=0
t=1
Neighborhood h
1
0
1
0
0
1
1
0
0
1
0
1
1
1
0
1
1
1
0
1
1
1
27
Rule 110 Space-Time Plot
28
Rule 30
current pattern
new state for center cell
111 110 101 100 011 010 001 000
0
0
0
1
1
1
1
0
29
Comments on Rule 30
 Generates apparent randomness, despite being finite
 Wolfram proposed using the central column as a pseudo-
random number generator
 Passes many tests for randomness, but many inputs produce
regular patterns:
 All zeroes
 00001000111000 repeated infinitely (try separating by 6 1s)
 Used in Mathematica for creating random integers (Wikipedia)
30
Variants
 Asynchronous CA

CA rules are typically applied simultaneously
across all cells in the lattice. This variant
allows the state of the cells to be updated
asynchronously.
 Probabilistic CA

The deterministic state-transitions are replaced
with specifications of the probabilities of the
cell-value assignments.
31
Variants
 Non-homogenous CA

State transition rules are allowed to vary from
cell to cell.
 Mobile CA



Some or all lattice sites are free to move about
the lattice.
Essentially primitive models of mobile robots.
Used to model some aspects of military
engagements.
32
Variants
 Structurally Dynamic CA

The topology (the sites and connections
among sites) are allowed to evolve.
33
Langton’s Lambda Parameter
 Wolfram’s classification scheme is phenomenological (argument
by visual inspection of space-time diagrams).
 Chris Langton (1986) quantified the classification scheme by
introducing the parameter  .
 Lambda is a statistic of the output states in the CA lookup table,
defined as the fraction of non-quiescent states in this table.
 The quiescent state is an arbitrarily chosen state
s
 Example: For a 2-state CA (   {0,1}), and quiescent state s=0,
lambda is the fraction of 1s in the output states of f
.
34
Lambda Parameter cont.

| h | nq
|h |
0    1.0 
1
k
 Where nq = the number of rules that map to the quiescent state.
 Using the lambda parameter to study CA s:
Table walkthrough: Start with one random table and progressively
perturb it through the range:
 Random table: Interpret lambda as a bias on the random selection
of states to fill up the table. (Get a new table for every new value
of lambda).
 Use various statistics to measure CA “average” behavior, as a function
of lambda:
 Single-site entropy.
 Same site mutual information across time steps.
 Two-site mutual information.

35
Density classification
•In the above example, we have assumed wrap-around,
and r=1.
•In this case, the CA has reached a ‘limit point’ from
which no escape is possible.
•CAs have been used for simulating fluid dynamics,
chemical oscillations, crystal growth, galaxy formation,
stellar accretion disks, fractal patterns on mollusc shells,
parallel formal language recognition, plant growth, traffic
flow, urban segregation, image processing tasks, etc …
36
Types of neighbourhood
Many more neighbourhood techniques
exist - see http://cell-auto.com and
follow the link to ‘neighbourhood
survey’
37
Wolfram’s CA Classification
 Class I: Eventually every cell in the array settles
into one state, never to change again.





Simple
Uniform final state (all black or all white)
Some examples are rules 0, 32, 128, 160, 250, 254
Analogous to computer programs that halt after a few
steps and to dynamical systems that have fixed-point
attractors.
after a finite number of time steps, the CA tends to achieve
a unique state from nearly all possible starting conditions
(limit points)
38
Wolfram Class I
39
Wolfram Class I
40
Wolfram’s CA Classification
 Class II: Eventually the array settles into a
periodic cycle of states (called a limit cycle).





Set of simple structures
Structures remain the same or repeat every so often
Examples include rules 132, 164, 218, 222
Analogous to computer programs that execute infinite
loops and to dynamical systems that fall into limit
cycles.
the CA creates patterns that repeat periodically or are
stable (limit cycles) – probably equivalent to a regular
grammar/finite state automaton
41
Wolfram’s Class II
42
Wolfram’s Class II
43
Wolfram’s CA Classification
 Class III: The array forms “aperiodic” random-like patterns.
 Appears random
 Smaller structures can be seen some at some level
 Most are expected to be computationally irreducible
 Examples include rules 22, 30, 126
 Analogous to computer programs that are pseudo-random number
generators (pass most tests for randomness, highly sensitive to
seed, or initial condition).
 Analogous to chaotic dynamical systems. Almost never repeat
themselves, sensitive to initial conditions, embedded unstable limit
cycles.
 from nearly all starting conditions, the CA leads to aperiodic-chaotic
patterns, where the statistical properties of these patterns are
almost identical (after a sufficient period of time) to the starting
patterns (self-similar fractal curves) – computes ‘irregular problems’
44
Wolfram’s Class III
45
Wolfram’s Class III
46
Wolfram’s Classification cont.
 Class IV: The array forms complex patterns with localized structure that
move through space and time:

Has order and randomness

Smaller scale structures interacting in complex ways
Examples include codes 1815, 2007, 1659, 2043
Recall: Codes are “totalistic” CAs where new color depends on average of
neighbors
Class 4 emerges as an intermediate class between classes 2 and 3
Difficult to describe. Not regular, not periodic, not random.
Speculate that it is interesting computation.
after a finite number of steps, the CA usually dies, but there are a few stable
(periodic) patterns possible (e.g. Game of Life) - Class 4 CA are believed to
be capable of universal computation






 Hypothesis: The most interesting and complex behavior occurs in Class
IV CA---the edge of chaos.
 Example: Rule 110
47
Wolfram’s Class IV
48
Wolfram’s Class IV
49
Exceptions
• Totalistic automata that don’t seem to fit
into just one class
• Codes 219, 438, 1380, 1632
50
Initial condition sensitivity
• Each class responds differently to a change
in its initial conditions
• Response types
• Class 1 changes always die out
• Changes continue on but are localized for
Class 2
• Uniform rate of change affecting the
whole system seen in Class 3
• Class 4 has nonuniform changes
51
Class 1
Class 2
52
Class 3
Class 4
53
Claim
• Differences in responses of classes show
each class handles information in a different
way
• Fundamental to our understanding of nature
54
Class 2
• Repetitive behavior
• No for support long-range communication
• Lack of long-range communication makes
systems of limited size forcing
repetitiveness
55
Observing systems of limited
behavior
• Limiting the size forces repetivness
• Period of repetition increases with size of
system
• With n cells, there are at most 2n possible
states (maximum period of 2n)
• Modulus
56
Repetition as a function of
system size
Rule 90
Rule 30
Rule 45
Rule 110
57
Class 3 randomness
• Randomness exists even without random
initial conditions
• Different initial conditions can produce
random behavior or nested pattern behavior
in the same rule (rule 22)
• Some rules need the random initial
condition to exhibit randomness (90) and
some rules don’t (30)
58
“Instrinsic Randomness”
• Do systems like rule 22 or rule 30 have
intrinsic randomness?
• Do these examples prove that certain
systems have intrinsic randomness and do
not depend on initial conditions?
• Special initial conditions can make class 3
systems behave like a class 2 or even a class
1 system (rule 126)
59
Rule 22 with
different initial
conditions
60
Rule 22 with
another set of
initial conditions
61
Rule 22
appearing
random with
different initial
conditions
62
Class 4 structures
• Certain structures will always last
• Any way to predict the structures of a given
rule and initial conditions?
• One can find all structures given a period,
but prediction is another matter
63
Attractors
• Sequences of cells restricted as iterations
progress, even with random initial
conditions
• Networks examples
64
Types of Networks
• Classes 1 and 2
• Never have more than t2 nodes after t
steps
• Classes 3 and 4
• Allowed sequences of cells becomes
more complicated
• Number of nodes increases at least
exponentially
65
Class 3 and 4 Exceptions
• Increase in network complexity not seen in
special initial conditions for rules 204, 240,
30, and 90
• Onto mappings defined
• Any other initial conditions than
“special” initial conditions rapidly
increase in complexity
66
Lambda Space and Wolfram Classes
0  c
1
1.0 
k
67
Lambda Parameter cont.
 Table walkthrough illustrates interesting change in dynamics as function
of lambda:
 Transient lengths increase dramatically in middle of the range (see
next slide).
 However, different traversals of lambda space using the
walkthrough method make the transition at different lambda values,
although there is a well defined distribution around a mean value.
 Size of the array has an effect on dynamics only for intermediate
values of lambda.
 Transient length depends exponentially on array size at lambda =
0.5.
 Overall evolutionary pattern in time is more random as lambda
increases past the transition region (use entropy and mutual
information).
 Transition region supports both static and propagating structures.
68
Comments on the Lambda Parameter
 Claims:
There is a phase transition between periodic and chaotic
behavior. Most complex behavior is in the vicinity of the
transition: The “edge of chaos.”
 CA s near the transition point correspond to Wolfram’s Class
IV.
 CA s capable of performing complex computations will be
found near the transition point (long transients).
 E.g., the game of life has lambda=0.273 (in the transition
region for K=2, N=9 2D CA s).
 Criticisms of the Lambda parameter:
 CA s with high lambda-value can still have simple behavior.
Lambda describes “average” behavior.
 Lambda does not take the initial state of the computation into
account (see Assignment 3).

69
John Conway’s Game of Life
 2D cellular automata system.
 Each cell has 8 neighbors - 4 adjacent
orthogonally, 4 adjacent diagonally. This is
called the Moore Neighborhood.
70
Simple rules, executed at each time
step:




A live cell with 2 or 3 live neighbors survives to
the next round.
A live cell with 4 or more neighbors dies of
overpopulation.
A live cell with 1 or 0 neighbors dies of
isolation.
An empty cell with exactly 3 neighbors
becomes a live cell in the next round.
71
Loops
 Assumptions
 Computation universality not required
 Characteristics
 8 states, 2D Cellular automata
 Needed CA grid of 100 cells
 Self Reproduction into identical copy
 Input tape with data and instructions
 Concept of Death
 Significance – Could be modeled through
computer programs
72
Langton’s Loop
0 – Background cell state
3, 5, 6 – Phases of reproduction
1 – Core cell state
4 – Turning arm left by 90 degrees
2 – Sheath cell state
state
7 – Arm extending forward cell state
73
Loop Reproduction
74
Loop Death
75
Langton’s Loops
Chris Langton formulated a much simpler form of self-rep
structure - Langton's loops - with only a few different states,
and only small starting structures.
76
Example:
Modelling Sharks and Fish:
77
 modeled a predator/prey relationship
 Begins with a randomly distributed population of fish,
sharks, and empty cells in a 1000x2000 cell grid (2
million cells)
 Initially,



50% of the cells are occupied by fish
25% are occupied by sharks
25% are empty
78
Here’s the number 2 million
Fish: red; sharks: yellow; empty: black 
79
Rules
A dozen or so rules describe life in each cell:
 birth, longevity and death of a fish or shark
 breeding of fish and sharks
 over- and under-population
 fish/shark interaction
 Important: what happens in each cell is determined
only by rules that apply locally, yet which often yield
long-term large-scale patterns.
80
Do a LOT of computation!
 Apply a dozen rules to each cell
 Do this for 2 million cells in the grid
 Do this for 20,000 generations
 Well over a trillion calculations per run!
 Do this as quickly as you can
81
Do a LOT of computation!
 We used a 20-CPU cluster in the Computer
Science Department (Galaxy)
 ‘Gal’ is the smaller of two clusters run by the
Department (larger one has 64 CPUs)
 15x faster than a single PC
 Longest runs still took about 45 minutes
 GO PARALLEL !!!
82
Rules in detail: Initial Conditions
Initially cells contain fish, sharks or are empty
 Empty cells = 0 (black pixel)
 Fish = 1 (red pixel)
 Sharks = –1 (yellow pixel)
83
Rules in detail: Breeding Rule
Breeding rule: if the current cell is empty
 If there are >= 4 neighbors of one species, and >= 3
of them are of breeding age,
 Fish breeding age >= 2,
 Shark breeding age >=3,
and there are <4 of the other species:
then create a species of that type
 +1= baby fish (age = 1 at birth)
 -1 = baby shark (age = |-1| at birth)
84
Breeding Rule: Before
EMPTY
85
Breeding Rule: After
86
Rules in Detail: Fish Rules
If the current cell contains a fish:
 Fish live for 10 generations
 If >=5 neighbors are sharks, fish dies (shark
food)
 If all 8 neighbors are fish, fish dies
(overpopulation)
 If a fish does not die, increment age
87
Rules in Detail: Shark Rules
If the current cell contains a shark:
 Sharks live for 20 generations
 If >=6 neighbors are sharks and fish
neighbors =0, the shark dies (starvation)
 A shark has a 1/32 (.031) chance of dying
due to random causes
 If a shark does not die, increment age
88
Shark Random Death: Before
I Sure Hope that the
random number
chosen is >.031
89
Shark Random Death: After
YES IT IS!!!
I LIVE 
90
Sample Code (C++): Breeding
91
Results
 Next several screens show behavior over a
span of 10,000+ generations
92
Generation: 0
93
Generation: 100
94
Generation: 500
95
Generation: 1,000
96
Generation: 2,000
97
Generation: 4,000
98
Generation: 8,000
99
Generation: 10,500
100
Long-term trends
 Borders tended to ‘harden’ along vertical,
horizontal and diagonal lines
 Borders of empty cells form between like
species
 Clumps of fish tend to coalesce and form
convex shapes or ‘communities’
101
Variations of Initial Conditions
 Still using randomly distributed populations:
 Medium-sized population. Fish/sharks
occupy:
1/16th of total grid
Fish: 62,703; Sharks: 31,301
 Very small population. Fish/sharks occupy:
1/800th of total grid
Initial population:
Fish: 1,298; Sharks: 609
102
Medium-sized population (1/16 of grid)
Generation 100
1000
4000
8000
2000
103
Very Small Populations
 Random placement of very small populations
can favor one species over another
 Fish favored: sharks die out
 Sharks favored: sharks predominate, but fish
survive in stable small numbers
104
Very Small Populations
Gen. 100
1500
4000
6000
8000
10,000
12,000
14,000
Ultimate welfare of sharks depends on initial random
placement of fish and sharks
105
Very small populations
 Fish can live in stable isolated communities
as small as 20-30
 A community of less than 200 sharks tends
not to be viable
106
This is what a community
of virtual plants looks like
Community image
Contrasting tones show patches
of resource depletion
107
This is a single propagule
of a virtual plant
It is about to grow in a
resource-rich above- and
below-ground environment
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
The plant has produced
abundant growth above- and
below-ground
and zones of resource
depletion have appeared
127
See Rod Hunt at
http://www.ex.ac.uk/~rh203/
for lots more about the plant life CA
and its uses
128